x + 6 Suppose we wish to give a 8-e proof that lim = -1. x-3 x+ – 4x3 + x2 + x + 6 х+6 a. Write +1 = (x – 3) · g(x). x4 – 4x3 + x2 + x + 6 b. Could we choose & = min ( 1, for some n E N? Explain. n c. If we choose 8 = min for some m E N, what is the smallest integer m that we could use? 4' m
x + 6 Suppose we wish to give a 8-e proof that lim = -1. x-3 x+ – 4x3 + x2 + x + 6 х+6 a. Write +1 = (x – 3) · g(x). x4 – 4x3 + x2 + x + 6 b. Could we choose & = min ( 1, for some n E N? Explain. n c. If we choose 8 = min for some m E N, what is the smallest integer m that we could use? 4' m
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. Suppose we wish to give a δ-ε proof that \( \lim_{{x \to 3}} \frac{x + 6}{x^4 - 4x^3 + x^2 + x + 6} = -1 \).
a. Write
\[
\frac{x + 6}{x^4 - 4x^3 + x^2 + x + 6} + 1 = (x - 3) \cdot g(x).
\]
b. Could we choose \( \delta = \min \left( 1, \frac{\varepsilon}{n} \right) \) for some \( n \in \mathbb{N} \)? Explain.
c. If we choose \( \delta = \min \left( \frac{1}{4}, \frac{\varepsilon}{m} \right) \) for some \( m \in \mathbb{N} \), what is the smallest integer \( m \) that we could use?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe35d795b-3b3a-45a5-a595-ba94ec33b836%2Fa838ec05-e78c-4f59-8a52-07fc9b083366%2Fgcmjuq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Suppose we wish to give a δ-ε proof that \( \lim_{{x \to 3}} \frac{x + 6}{x^4 - 4x^3 + x^2 + x + 6} = -1 \).
a. Write
\[
\frac{x + 6}{x^4 - 4x^3 + x^2 + x + 6} + 1 = (x - 3) \cdot g(x).
\]
b. Could we choose \( \delta = \min \left( 1, \frac{\varepsilon}{n} \right) \) for some \( n \in \mathbb{N} \)? Explain.
c. If we choose \( \delta = \min \left( \frac{1}{4}, \frac{\varepsilon}{m} \right) \) for some \( m \in \mathbb{N} \), what is the smallest integer \( m \) that we could use?
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